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				<span style="position: absolute;left:15px;bottom:15px;width:90%;"><font class="view-text" style="color:#fcfcfc;font-size:25px">题解 CF891E 【Lust】</font><br><a href="/tags/2021/" class="tag"><span  style="background-color: rgb(52, 152, 219);">2021</span></a>&nbsp;<a href="/tags/生成函数/" class="tag"><span  style="background-color: rgb(231, 76, 60);">生成函数</span></a>&nbsp;<a href="/tags/题解/" class="tag"><span  style="background-color: rgb(82, 196, 26);">题解</span></a></span>
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                <h2 id="_1">题意</h2>
<p><input value="在洛谷上查看" type="button" onclick="creat('https://www.luogu.com.cn/problem/CF891E')" class="btn btn-link"/></p>
<p>有 <script type="math/tex">n</script> 个数 <script type="math/tex">a_1,\ldots,a_n</script>，进行 <script type="math/tex">k</script> 次操作，每次随机选一个 <script type="math/tex">x\in[1,n]</script> 减 <script type="math/tex">1</script>，并将答案加上除 <script type="math/tex">a_x</script> 外的所有数的乘积。</p>
<h2 id="_2">题解</h2>
<p>
<script type="math/tex; mode=display">\color{orange}\Huge\text{降智人是人下人}</script>
</p>
<hr>
<p>大意了这都想不出来！！</p>
<p>考虑操作一个 <script type="math/tex">x</script> 对答案的贡献：
<script type="math/tex; mode=display">\prod_{i\neq x}a_i=\prod_{i=1}^na_i-(a_x-1)\prod_{i\neq x}a_i</script>
就是将前一个的所有数的积减去操作后的积的差分操作。所以最终的答案就是一开始所有数的乘积减去最后一个的乘积。</p>
<p>前者很好算，来考虑后者，不难发现就是：
<script type="math/tex; mode=display">k![z^k]\prod_{i=1}\sum_{j=0}\frac{a_i-j}{j!}z^j=k![z^k]e^{nz}\prod_{i=1}^n(a_i-z)</script>
<script type="math/tex">n\le 5000</script> 随便搞一下就好了。</p>
<h2 id="_3">代码</h2>
<div class="highlight"><pre><span></span><code><span class="linenos" data-linenos=" 1 "></span><span class="cp">#include</span><span class="cpf">&lt;bits/stdc++.h&gt;</span><span class="cp"></span>
<span class="linenos" data-linenos=" 2 "></span><span class="k">using</span> <span class="k">namespace</span> <span class="n">std</span><span class="p">;</span>
<span class="linenos" data-linenos=" 3 "></span><span class="k">const</span> <span class="kt">int</span> <span class="n">mod</span><span class="o">=</span><span class="mf">1e9</span><span class="o">+</span><span class="mi">7</span><span class="p">;</span>
<span class="linenos" data-linenos=" 4 "></span><span class="k">struct</span> <span class="nc">modint</span><span class="p">{</span>
<span class="linenos" data-linenos=" 5 "></span>    <span class="kt">int</span> <span class="n">x</span><span class="p">;</span>
<span class="linenos" data-linenos=" 6 "></span>    <span class="n">modint</span><span class="p">(</span><span class="kt">int</span> <span class="n">o</span><span class="o">=</span><span class="mi">0</span><span class="p">){</span><span class="n">x</span><span class="o">=</span><span class="n">o</span><span class="p">;}</span>
<span class="linenos" data-linenos=" 7 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">=</span> <span class="p">(</span><span class="kt">int</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="n">o</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos=" 8 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">+=</span><span class="p">(</span><span class="n">modint</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="o">+</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="o">&gt;=</span><span class="n">mod</span><span class="o">?</span><span class="n">x</span><span class="o">+</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="o">-</span><span class="nl">mod</span><span class="p">:</span><span class="n">x</span><span class="o">+</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos=" 9 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">-=</span><span class="p">(</span><span class="n">modint</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="o">-</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="o">&lt;</span><span class="mi">0</span><span class="o">?</span><span class="n">x</span><span class="o">-</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="o">+</span><span class="nl">mod</span><span class="p">:</span><span class="n">x</span><span class="o">-</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos="10 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">*=</span><span class="p">(</span><span class="n">modint</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="mf">1l</span><span class="n">l</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">o</span><span class="p">.</span><span class="n">x</span><span class="o">%</span><span class="n">mod</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos="11 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">^=</span><span class="p">(</span><span class="kt">int</span> <span class="n">b</span><span class="p">){</span>
<span class="linenos" data-linenos="12 "></span>        <span class="n">modint</span> <span class="n">a</span><span class="o">=*</span><span class="k">this</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos="13 "></span>        <span class="k">for</span><span class="p">(;</span><span class="n">b</span><span class="p">;</span><span class="n">b</span><span class="o">&gt;&gt;=</span><span class="mi">1</span><span class="p">,</span><span class="n">a</span><span class="o">*=</span><span class="n">a</span><span class="p">)</span><span class="k">if</span><span class="p">(</span><span class="n">b</span><span class="o">&amp;</span><span class="mi">1</span><span class="p">)</span><span class="n">c</span><span class="o">*=</span><span class="n">a</span><span class="p">;</span>
<span class="linenos" data-linenos="14 "></span>        <span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="n">c</span><span class="p">.</span><span class="n">x</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;</span>
<span class="linenos" data-linenos="15 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos="16 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">/=</span><span class="p">(</span><span class="n">modint</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="o">*</span><span class="k">this</span> <span class="o">*=</span><span class="n">o</span><span class="o">^=</span><span class="n">mod</span><span class="mi">-2</span><span class="p">;}</span>
<span class="linenos" data-linenos="17 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">+=</span><span class="p">(</span><span class="kt">int</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="o">+</span><span class="n">o</span><span class="o">&gt;=</span><span class="n">mod</span><span class="o">?</span><span class="n">x</span><span class="o">+</span><span class="n">o</span><span class="o">-</span><span class="nl">mod</span><span class="p">:</span><span class="n">x</span><span class="o">+</span><span class="n">o</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos="18 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">-=</span><span class="p">(</span><span class="kt">int</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="o">-</span><span class="n">o</span><span class="o">&lt;</span><span class="mi">0</span><span class="o">?</span><span class="n">x</span><span class="o">-</span><span class="n">o</span><span class="o">+</span><span class="nl">mod</span><span class="p">:</span><span class="n">x</span><span class="o">-</span><span class="n">o</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos="19 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">*=</span><span class="p">(</span><span class="kt">int</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="n">x</span><span class="o">=</span><span class="mf">1l</span><span class="n">l</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">o</span><span class="o">%</span><span class="n">mod</span><span class="p">,</span><span class="o">*</span><span class="k">this</span><span class="p">;}</span>
<span class="linenos" data-linenos="20 "></span>    <span class="n">modint</span> <span class="o">&amp;</span><span class="k">operator</span> <span class="o">/=</span><span class="p">(</span><span class="kt">int</span> <span class="n">o</span><span class="p">){</span><span class="k">return</span> <span class="o">*</span><span class="k">this</span> <span class="o">*=</span> <span class="p">((</span><span class="n">modint</span><span class="p">(</span><span class="n">o</span><span class="p">))</span><span class="o">^=</span><span class="n">mod</span><span class="mi">-2</span><span class="p">);}</span>
<span class="linenos" data-linenos="21 "></span>    <span class="k">template</span><span class="o">&lt;</span><span class="k">class</span> <span class="nc">I</span><span class="o">&gt;</span><span class="k">friend</span> <span class="n">modint</span> <span class="k">operator</span> <span class="o">+</span><span class="p">(</span><span class="n">modint</span> <span class="n">a</span><span class="p">,</span><span class="n">I</span> <span class="n">b</span><span class="p">){</span><span class="k">return</span> <span class="n">a</span><span class="o">+=</span><span class="n">b</span><span class="p">;}</span>
<span class="linenos" data-linenos="22 "></span>    <span class="k">template</span><span class="o">&lt;</span><span class="k">class</span> <span class="nc">I</span><span class="o">&gt;</span><span class="k">friend</span> <span class="n">modint</span> <span class="k">operator</span> <span class="o">-</span><span class="p">(</span><span class="n">modint</span> <span class="n">a</span><span class="p">,</span><span class="n">I</span> <span class="n">b</span><span class="p">){</span><span class="k">return</span> <span class="n">a</span><span class="o">-=</span><span class="n">b</span><span class="p">;}</span>
<span class="linenos" data-linenos="23 "></span>    <span class="k">template</span><span class="o">&lt;</span><span class="k">class</span> <span class="nc">I</span><span class="o">&gt;</span><span class="k">friend</span> <span class="n">modint</span> <span class="k">operator</span> <span class="o">*</span><span class="p">(</span><span class="n">modint</span> <span class="n">a</span><span class="p">,</span><span class="n">I</span> <span class="n">b</span><span class="p">){</span><span class="k">return</span> <span class="n">a</span><span class="o">*=</span><span class="n">b</span><span class="p">;}</span>
<span class="linenos" data-linenos="24 "></span>    <span class="k">template</span><span class="o">&lt;</span><span class="k">class</span> <span class="nc">I</span><span class="o">&gt;</span><span class="k">friend</span> <span class="n">modint</span> <span class="k">operator</span> <span class="o">/</span><span class="p">(</span><span class="n">modint</span> <span class="n">a</span><span class="p">,</span><span class="n">I</span> <span class="n">b</span><span class="p">){</span><span class="k">return</span> <span class="n">a</span><span class="o">/=</span><span class="n">b</span><span class="p">;}</span>
<span class="linenos" data-linenos="25 "></span>    <span class="k">friend</span> <span class="n">modint</span> <span class="k">operator</span> <span class="o">^</span><span class="p">(</span><span class="n">modint</span> <span class="n">a</span><span class="p">,</span><span class="kt">int</span> <span class="n">b</span><span class="p">){</span><span class="k">return</span> <span class="n">a</span><span class="o">^=</span><span class="n">b</span><span class="p">;}</span>
<span class="linenos" data-linenos="26 "></span>    <span class="n">modint</span> <span class="k">operator</span> <span class="o">-</span> <span class="p">()</span> <span class="p">{</span><span class="k">return</span> <span class="n">x</span><span class="o">?</span><span class="n">mod</span><span class="o">-</span><span class="nl">x</span><span class="p">:</span><span class="mi">0</span><span class="p">;}</span>
<span class="linenos" data-linenos="27 "></span><span class="p">};</span>
<span class="linenos" data-linenos="28 "></span><span class="k">const</span> <span class="kt">int</span> <span class="n">N</span><span class="o">=</span><span class="mi">5010</span><span class="p">;</span>
<span class="linenos" data-linenos="29 "></span><span class="kt">int</span> <span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">;</span><span class="n">modint</span> <span class="n">a</span><span class="p">[</span><span class="n">N</span><span class="p">];</span>
<span class="linenos" data-linenos="30 "></span><span class="n">modint</span> <span class="n">poly</span><span class="p">[</span><span class="n">N</span><span class="p">];</span>
<span class="linenos" data-linenos="31 "></span><span class="kt">signed</span> <span class="nf">main</span><span class="p">(){</span>
<span class="linenos" data-linenos="32 "></span>    <span class="n">scanf</span><span class="p">(</span><span class="s">&quot;%d%d&quot;</span><span class="p">,</span><span class="o">&amp;</span><span class="n">n</span><span class="p">,</span><span class="o">&amp;</span><span class="n">k</span><span class="p">);</span>
<span class="linenos" data-linenos="33 "></span>    <span class="n">modint</span> <span class="n">tmp</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">res</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">inv</span><span class="o">=</span><span class="n">modint</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">n</span><span class="p">;</span>
<span class="linenos" data-linenos="34 "></span>    <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span><span class="n">i</span><span class="o">&lt;=</span><span class="n">n</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
<span class="linenos" data-linenos="35 "></span>        <span class="n">scanf</span><span class="p">(</span><span class="s">&quot;%d&quot;</span><span class="p">,</span><span class="o">&amp;</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span><span class="n">res</span><span class="o">*=</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
<span class="linenos" data-linenos="36 "></span>    <span class="n">poly</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos="37 "></span>    <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span><span class="n">i</span><span class="o">&lt;=</span><span class="n">n</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">){</span>
<span class="linenos" data-linenos="38 "></span>        <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">j</span><span class="o">=</span><span class="n">n</span><span class="p">;</span><span class="n">j</span><span class="p">;</span><span class="n">j</span><span class="o">--</span><span class="p">)</span>
<span class="linenos" data-linenos="39 "></span>            <span class="n">poly</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">=-</span><span class="n">poly</span><span class="p">[</span><span class="n">j</span><span class="mi">-1</span><span class="p">]</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">poly</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<span class="linenos" data-linenos="40 "></span>        <span class="n">poly</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*=</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
<span class="linenos" data-linenos="41 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos="42 "></span>    <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="n">k</span><span class="p">;</span><span class="n">i</span><span class="o">&gt;=</span><span class="n">k</span><span class="o">-</span><span class="n">n</span><span class="p">;</span><span class="n">i</span><span class="o">--</span><span class="p">){</span>
<span class="linenos" data-linenos="43 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="n">i</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">)</span><span class="k">continue</span><span class="p">;</span>
<span class="linenos" data-linenos="44 "></span>        <span class="n">res</span><span class="o">-=</span><span class="n">tmp</span><span class="o">*</span><span class="n">poly</span><span class="p">[</span><span class="n">k</span><span class="o">-</span><span class="n">i</span><span class="p">];</span>
<span class="linenos" data-linenos="45 "></span>        <span class="n">tmp</span><span class="o">*=</span><span class="n">i</span><span class="p">;</span><span class="n">tmp</span><span class="o">*=</span><span class="n">inv</span><span class="p">;</span>
<span class="linenos" data-linenos="46 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos="47 "></span>    <span class="n">printf</span><span class="p">(</span><span class="s">&quot;%d&quot;</span><span class="p">,</span><span class="n">res</span><span class="p">);</span>
<span class="linenos" data-linenos="48 "></span><span class="p">}</span>
</code></pre></div>
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